An $\widetilde{O} (n^{3/7})$ Round Parallel Algorithm for Matroid Bases

2026-05-05

AI Summary

This work investigates the minimum number of rounds required for parallel adaptive algorithms to find a basis of an $n$-element matroid in the independence oracle model. By introducing a novel algorithmic framework that shifts the analytical perspective from individual elements to joint dependency structures among multiple elements, and by integrating dependency analysis from randomized circuits with adaptive parallel query strategies, the study breaks through a complexity barrier that has persisted for nearly four decades. The proposed method achieves a round complexity of $\widetilde{O}(n^{3/7})$ while submitting only a polynomial number of independence queries per round, significantly improving upon the previous best-known bound of $\widetilde{O}(n^{7/15})$.

Abstract

We study the parallel (adaptive) complexity of the classic problem of finding a basis in an $n$-element matroid, given access via an \emph{independence oracle}. In this model, the algorithm may submit polynomially many independence queries in each round, and the central question is: how many rounds are necessary and sufficient to find a basis? Karp, Upfal, and Wigderson (FOCS~1985, JCSS~1988; hereafter KUW) initiated this study, showing that $O(\sqrt{n})$ adaptive rounds suffice for any matroid, and that $\widetildeΩ(n^{1/3})$ rounds are necessary even for partition matroids. This left a substantial gap that persisted for nearly four decades, until Khanna, Putterman, and Song (FOCS~2025; hereafter KPS) achieved $\widetilde O(n^{7/15})$ rounds, the first improvement since~KUW. In this work, we make another conceptual advance beyond KPS, giving a new algorithm that finds a matroid basis in $\widetilde O(n^{3/7})$ rounds. We develop a structural and algorithmic framework that brings a new lens to the analysis of random circuits, moving from reasoning about individual elements to understanding how dependencies span multiple elements simultaneously.

Problem

• matroid basis
• parallel algorithm
• adaptive complexity
• independence oracle
• round complexity

Innovation

• matroid basis
• parallel algorithm
• adaptive complexity
• independence oracle
• random circuits